On the regularity analysis of interpolatory Hermite subdivision schemes

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:302

Article

Yu, TPY

关键词: holder regularity;    subdivision scheme;    refinement equation;    hermite interpolation;    wavelets;    multiwavelets;    joint spectral radii;    matrix polynomial;    positivity;    Fejer-Riesz factorization;   

DOI  :  10.1016/j.jmaa.2004.07.051

来源: Elsevier

【 摘 要 】

It is well known that the critical Holder regularity of a subdivision schemes can typically be expressed in terms of the joint-spectral radius (JSR) of two operators restricted to a common finite-dimensional invariant subspace. In this article, we investigate interpolatory Hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders. The latter include as special cases the well-known Lagrange interpolatory subdivision schemes by Deslauriers and Dubuc. We first show how to express the critical Holder regularity of such a scheme in terms of the joint-spectral radius of a matrix pair {F-0, F-1} given in a very explicit form. While the so-called finiteness conjecture for JSR is known to be not true in general, we conjecture that for such matrix pairs arising from Hermite interpolatory schemes of optimal accuracy orders a strong finiteness conjecture holds: rho(F-0, F-1) = rho(F-0) = rho(F-1). We prove that this conjecture is a consequence of another conjectured property of Hermite interpolatory schemes which, in turn, is connected to a kind of positivity property of matrix polynomials. We also prove these conjectures in certain new cases using both time and frequency domain arguments; our study here strongly suggests the existence of a notion of positive definiteness for non-Hermitian matrices. (C) 2004 Elsevier Inc. All rights reserved.

【 授权许可】

Free